Probabilistic And Stochastic UML Statecharts - Software Modeling ...

Probabilistic And Stochastic 

UML Statecharts 

Speaker: Abdallah Salim 

Tutor: Dr. David N. Jansen 

Chair for Software Modeling and Verification 

RWTH Aachen, Germany 

SS 2007

Overview 

An introduction to Statecharts 

StoCharts 

Syntax 

Semantics 

Case Study: A Beverages dispenser 

Conclusion 

2

The UML perspective - A scenario 

Name: Mr Troy aka Mr T 

Age: 29 

Position: Software Engineer 

T 

T seems to have an idea ! 

3 

Name: Mr McNamara aka Mr M 

Age: 27 

Position: Visual designer 

M

The UML perspective - Getting the message across 

Mr. T has a great 

design idea 

What do 

you think 

M ? 

T M 

*** 

but ... 

4 

Mr. M doesn’t seem to 

understand T’s design 

What is 

that T ?

Mr. T has an idea in 

his head 

*** 

The UML perspective - The Problem 

T 

*** 

but ... 

5 

Mr. M forms a 

completely different 

one in his head 

M 

ooo 

This is ... 

NOT good

Mr. T redesigns with 

UML.... 

UML 

The UML perspective - Enter UML 

T 

Ok what 

about this ? 


Why does Mr M get 

it ? 

UML provides a 

Lingua franca that 

promotes 

communication 

between designers 

Lingua franca = 

Common Language 

6 

I see it 

now ! 

Mr. M gets it !!! 

M 

UML

Introducing StoCharts - An Intuition 

StoChart Example: 6 wins the game 

5/6 / player.not six 

Rolling 

End of roll 

P 

Won 

after(EXP[3sec]) 

/ player.stop rolling 

1/6 / player.six 

7 

(1) Dice starts rolling 

(takes some time to stop: 

3 sec) 

(2) Nondeterministic choice 

- Dice cannot be read 

- Dice can be read 

(3) Probabilistic choice 

- {1,2,3,4,5} face-up 

(start rolling again) 

- 6 face-up 

(game over) 

⊥ 

: No need to wait to take edge

=> 

Some Distinctions - Probabilistic choice 

(Example: Throwing a fair coin) 

Probabilistic choice: 

It is not clear which possibility out of a discrete set a 

system may take. 

So what does “not clear” mean ? 

- Prediction: 50% of the time heads/tails will 

occur. 

- BUT impossible to predict the actual outcome for 

each toss i.e., whether it’s a head or a tail. 

8

=> 

Some Distinctions - Stochastic timing 

(Example: Video Streaming) 

Client 

Video displayed 

Video request 

Stream 

buffered 

Stochastic timing ? 

- Some time needed to schedule video packets (server). 

- Some time for video to be displayed (client). 

9 

Webserver 

Scheduling of video 

packets

Some Distinctions - Probabilistic vs (non) 

deterministic Phenomena 

Probabilistic (Examples: Weather predictions - Probability of 

rain, snow etc; Rolling of dice in a board game) 

Deterministic (Example: Forced moves in chess) 

Nondeterministic (Examples: Free moves in chess and Tic Tac 

Toe) 

10

Some Distinctions - Non/Not determinism 

Nondeterministic Not determinstic 

Not deterministic Phenomena 

Only infers that a phenomenon is either probabilistic or 

nondeterministic (and nothing more !!) 

An unfortunate terminology but 

one we have to live with :-( 

11

Some Distinctions - System vs Environment randomness 

Environment randomness 

Environment randomness (Example: Reactive system - Bank 

ATM, Dispenser, etc) 

Intuition: System exposed to external random stimuli and 

reacts. 

System reaction is NOT random 

12

Some Distinctions - System randomness 

Requirement : Modems must transmit on different frequencies 

modem 1 modem 2 

Notation: 

- Hj : “Heads” for modem j (1

Overview 


StoCharts 

Syntax 

Semantics 


Conclusion 

14

=> 

=> 

UML Extensions - P-Statecharts & StoCharts 

We can extend UML Statecharts: 2 fold Extension 

Extension 1: Probabilistic choice 

(Example: Heads/Tails choice on a coin toss) 

Extension 2: Stochastic timing 

(Example: Timing in a Drink dispenser) 

Statecharts + Extension 1 

= Probabilistic Statecharts (aka P-Statecharts) 

Statecharts + Extension 1 + Extension 2 

= Stochastic Statecharts (aka StoCharts) 

15

UML Extensions - Random probability 

distributions 

DET [x min] 

A deterministic distribution with a duration of x minutes 

EXP [x min] 

A negative exponential distribution with a mean of x minutes 

UNIF [x min, y min] 

A uniform distribution in the interval from x to y minutes 

16

1.00 

0.75 

0.50 

0.25 

0 

UML Extensions - Random probability 

Prob. DET[5min] 

Discrete 

function 

0 1 2 3 4 5 6 7 8 9 10 11 12 

Prob. 

1.00 

0.75 

0.50 

0.25 

0 

Time [min] 

17 

Prob. 

1.00 

0.75 

0.50 

0.25 

0 

UNIF[5min,10min] 

0 1 2 3 4 5 6 7 8 9 10 11 12 

EXP[1min] 

0 1 2 3 4 5 6 7 8 9 10 11 12 

Time [min] 

Time [min]

UML Extensions - Statechart relations 

In a nutshell ... 

UML Statecharts are 

trivial P-Statecharts 

UML Statecharts + 

Probabilistic choice => 

P-Statecharts 

StoCharts 

P-Statecharts 


Statecharts 

18 

UML Statecharts are 

trivial StoCharts 

P-Statecharts + 

Stochastic timing => 

StoCharts

Overview 


StoCharts 

Syntax 

Semantics 


Conclusion 

19

A syntactical perspective - StoChart drawing 

Nodes 

C 

A 

B A 

Parent node C encloses its 

children A, B 

(Default node: A) 

20 

A C 

B E D 

F (AND) node partitioned by (OR) 

children A and C 

(Default nodes: B,D) 

F


Trivial P-edges 

X e[g]/A Y 

Verifying 

User 

X : Set of source nodes 

Y : set of target nodes 

e : event 

g : guard 

A : action set 

Here: Actions “A” have the form (send) j.e (send event e to the 

external component identified by j) 

Example (Router Login) 

Check password[password=”admin”] 

/User.Logged in 

21 

Showing 

Admin Page


Non trivial P-edges 

e[g]/ 

X P 

Example (File downloading) 

Checking 

origin 

p/A 

Check country/ 

. 

. 

. 

22 

R 

p’/A S p=p’ is also possible 

0.4/downloader.UK 

servers available 

P 

0.6/ 

downloader.USA 

servers available 

X : Set of source nodes 

Y : set of target nodes 

e : event 

g : guard 

A : action set 

p,p’: Probabilities 

Showing 

UK 

Server 1 

Showing 

USA Server 

7

A syntactical perspective - System Perspective 

- Intuition: A system is a finite collection of 

intercommunicating StoCharts 

- Formally: Denotable by the set seen below. 

{SC i |1 ≤ i ≤ n,n ∈N} 

23

StoChart 4-Tuple 

Nodes i 

Events i 

Vars i 

PEdges i 

A syntactical perspective - StoChart 

SC i = (Nodes i,Events i,Vars i,PEdges i ) 

A finite set of nodes structured in a tree 

A finite set of events 

A finite set of local variables 

A finite set of P-edges 

24

A syntactical perspective - StoChart Events 


A finite set of events (Signals that may trigger a state transition) 

pseudo event after(F) - stochastic random delays 

⊥∉Events i 

PSEvents i 

No event is required to trigger a P-edge 

Set of Pseudo events 

25

A syntactical perspective - StoChart Variables 


- Intuition: A finite set of local variables with an initial 

value that assigns initial values to variables 

- Formally: Denotable by the function below 

V 0 :Vars i → Z 

26

A syntactical perspective - StoChart P-edges 

P-edge 4-Tuple (X,e, g, P) 

A finite set of P-edges 

X ⊆ Nodes i 

e ∈Events i ∪ PSEvents i ∪ {⊥} 

g ∈Guards i 

A non-empty set of source state nodes 

27 

Triggering event that occurs 

A guard existing on the edge 

P denotes the possible actions and target state nodes (formal 

description needed)

A syntactical perspective - StoChart P-edges 

A formal description of P 

(X,e, g, P) 

P can be formally described by a probability measure in a discrete 

probability space 

Pow 

(Pow(Actions i ) × (Pow(Nodes i ) − {∅}), P) 

: Denotes a Power set 

28 

Ω 

Denoting the set of possible outcomes

A syntactical perspective - StoChart scopes 

Intuition: The scope of a P-edge is the smallest (in the parent-child 

hierarchy) OR-node containing both the source nodes and the target 

nodes 

H 

F 

E D 

A B 

C 

G 

29 

Scope(A->H) = root 

Scope(A->D) = G 

Scope(A->B) = C

Overview 


StoCharts 

Syntax 

Semantics 


Conclusion 

30

IOSA 7-Tuple (L,l 0 ,T , A, I,O,Δ) 

L 

T 

A 

A semantical perspective - Automaton approach 

I 

O 

Δ 

A set L of locations with an initial location 

A set of timers (each possessing a cdf) 

An input transition relation 

A probabilistic output transition 

relation 

A delay transition relation 

31 

IOSA = 

Stochastic I/O- 

Automaton 

l 0 ∈L 

A set of actions partitioned into input-actions 

(from external environment) and output-actions (system intrinsic)



I 

An input transition relation 

I : L × A in → L 

32



O A probabilistic output transition relation 

O ⊆ L × P(A out × Pow(T ) × L) 

33



Δ 

A delay transition relation 

Δ ⊆ L × T × L 

34

A semantical perspective - Enabled Transitions 

IOSA 7-Tuple 

Transition 

o 

i 

d 

(L,l 0 ,T , A, I,O,Δ) 

35 

Enabled 

Always 

Only if Input action present 

(Input enabledness) 

Only if 

timer expires (reaches zero) 

o: output transition 

i: input transition 

d: delay transition

ATM request 

allow or deny 

withdraw 

A semantical perspective - IOSA example 

-> i 

i 

Bank 

o: output transition, i: input transition 

d: delay transition 

output transitions always enabled ! 

input transition enabled 

i:request 

Idle 

i 

:request 

timer set timer has expired 

1 

Preparing o: set t Checking d:t 

Check 

t 

to check 

finished 

timeout ? 

i:withdraw 

i:withdraw 

Error 

i:withdraw 

i:withdraw 

o: 

No withdraw without 

positive verification ! 

i:withdraw 

36 

Bank behavior being modeled ! 

0.02, deny 

Probabilistic choice 

0.98, allow 

Check 

positive

Overview 


StoCharts 

QoS Extensions 

Syntax 

Semantics 


Conclusion 

37

How does it work ? 

Drinks offered ? 

“Espresso” coffee and 

“oolong” tea. 

Payment ? 

A beverage dispenser - Intuition 

Money chips - 1 Chip 

per drink. 

38

A beverage dispenser - Modeling perspective 

What are we modeling ? 

The behavior of the system (system randomness), as seen 

by the customer. 

What we are not modeling 

The expected/desired behavior of the customer 

(environmental randomness) is NOT being considered here. 

39

A naive UML Statechart - Events 

Events are sent from the 

customer to the HBD. 

insert chip. 

select espresso. 

select oolong. 

HBD = Hot Beverages Dispenser 

40

select Espresso/ 

customer.brew 

espresso 

A naive UML Statechart - Closer look 

Waiting 

for chip 

insert chip/ 

Waiting 

for beverage selection 

HBD - Hot 

Beverages 

Dispenser 

select Oolong/ 

customer.brew 

oolong 

41 

insert chip/: 

Intuition: In the event of a 

chip being inserted, the 

system evolves to its next 

state 

select oolong/customer.brew 

oolong: 

Intuition: In the event that 

oolong is selected, send the 

event “brew oolong” to the 

customer 

Analog for espresso

An improved UML Statechart - Closer look 

What’s wrong with our naive design ? 

Incomplete - Drinking 

what we paid for ? 

Completing the model 

“Placing” of cups in 

cup holder 

“Filling” of the cups 

“waiting” for 

someone (not a 

system) to remove 

the cup 

42 

remove cup/ 


customer.brew 

espresso 

Filling cup 

with espresso 

cup full/customer. 

notify drink ready 

Waiting 

for chip 

Placing cup 

insert chip/ 

Waiting for cup 

to be removed 

Improved 

HBD (IHBD) 


customer.brew 

oolong 

Filling cup 

with oolong 


notify drink ready

Extending UML Statecharts - UML Limitations 

Limitations of the IHBD 

Modeling system 

requirements ? 

(1) No cups => coin 

returned to customer 

(2) Return a chip 

exactly after 10 seconds 

(“timeout”) 

(3) Brewing takes about 

5 seconds (“timeout”) 

Realisation: Impossible to model our 

requirements without extensions !!! 

43 

remove cup/ 


customer.brew 

espresso 

Filling cup 

with espresso 



Waiting 

for chip 

Placing cup 

insert chip/ 


to be removed 

Improved 

HBD (IHBD) 


customer.brew 

oolong 

Filling cup 

with oolong 


notify drink ready

Extending UML Statecharts - Modeling requirements 

Requirements to be modeled 

(1) No cups => coin returned 

to customer 

(2) Return a chip exactly 

after 10 seconds 


(3) Brewing takes about 5 

seconds (“timeout”) 

44 

1 

2 

3 

IHBD extended 

(probabilistic choice & 

stochastic timing) 

“Probabilistic choice” 

Enough cups vs not enough cups 

“Stochastic timing” 

after(DET [10s] ) / 

customer.return chip 

“Stochastic timing” 

after(EXP [5s] ) / 

customer.notify drink ready

Extending UML Statecharts - Birth of a StoChart 

Requirements to be modeled 

(1) No cups => coin 

returned to customer 

(2) Return a chip 

exactly after 10 

seconds (“timeout”) 

(3) Brewing takes 

about 5 seconds 


(1) 

(2) 

(3) 

45 

Stochastic IHBD (S-IHBD) 


customer.brew 

espresso 

0.8/ 

Filling cup 

with espresso 

after (EXP[5 s])/ 

customer. 


Waiting 

for chip 

Placing cup 

insert chip/ 


to be removed 

0.2/ 

P 

Out of cups 

after(DET[10s] / 



customer.brew 

oolong 

Filling cup 

with oolong 


customer. 


remove cup/ 

insert chip/ 

customer.return chip

emove cup/ 

Extending UML Statecharts - Statecharts compared 


customer.brew 

espresso 

Filling cup 

with espresso 

Waiting 

for chip 

Placing cup 

insert chip/ 




to be removed 


customer.brew 

oolong 

Filling cup 

with oolong 

IHBD - without extensions. 

Modeling requirements 

involving Stochastic behavior 

not possible ! 

Probabilistic 

choice 

46 


customer.brew 

espresso 

0.8/ 

Filling cup 

with espresso 


customer. 


Waiting 

for chip 

Placing cup 

insert chip/ 


to be removed 

0.2/ 

P 

Out of cups 

after(DET[10s] / 



customer.brew 

oolong 

Filling cup 

with oolong 


customer. 


remove cup/ 

insert chip/ 


Stochastic 

timing 

S-IHBD (IHBD with 

extensions). 

Modeling requirements 

involving Stochastic behavior 

now becomes possible !!

Overview 


StoCharts 

QoS Extensions 

Syntax 

Semantics 


Conclusion 

47

What have we learnt ? 

- Awareness of real UML limitations and the need to extend them. 

- Extending UML Statecharts with : 

- Probabilistic choice (P-Statecharts). 

- Stochastic timing (StoCharts). 

- Informal and formal syntax of StoCharts ( with examples ). 

- Informal semantics : Stochastic Input/Output Automata (IOSA). 

- Application of UML extensions: Case study of beverage dispenser. 

48

Some useful reading 

Martin Fowler. UML Distilled: A Brief Guide to the 

standard Object Modeling Language, Third edition. 

Addison-Wesley Professional, 3rd edition, 2003. 

David N. Jansen. Extensions of Statecharts with 

Probability, Time, and Stochastic Timing. PhD thesis, 

Universiteit Twente, Bern, 2003. 

Scott Kim. Interdisciplinary cooperation. In Brenda Laurel, 

editor, the Art of Human Computer Interface Design, 

pages 31-44, Addison Wesley, 1990. 

49

Thank you 

for your attention !

Probabilistic And Stochastic UML Statecharts - Software Modeling ...

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